The stellar mass spectrum from non-isothermal gravoturbulent fragmentation

0.4 MB·English

by  Anne-Katharina Jappsen

Astronomy&Astrophysicsmanuscriptno.akjappsen-v3 February2,2008
(DOI:willbeinsertedbyhandlater)
The stellar mass spectrum from non-isothermal gravoturbulent
fragmentation
Anne-Katharina Jappsen1,RalfS.Klessen1,RichardB.Larson2,YuexingLi3,4,andMordecai-Mark MacLow3,4
1 AstrophysikalischesInstitutPotsdam,AnderSternwarte16,D-14482Potsdam,Germany
5 2 DepartmentofAstronomy,YaleUniversity,NewHaven,CT06520-8101,USA
0
3 DepartmentofAstrophysics,AmericanMuseumofNaturalHistory,79thStreetatCentralParkWest,NewYork,NY10024-
0
5192,USA
2
4 DepartmentofAstronomy,ColumbiaUniversity,NewYork,NY10027,USA

n
a
Received<date>/Accepted<date>
J

1
Abstract. Identifyingtheprocessesthatdeterminetheinitialmassfunctionofstars(IMF)isafundamentalprobleminstar
1
formation theory. One of themajor uncertainties istheexact chemical stateof thestar forming gasand itsinfluence onthe

2 dynamicalevolution.Mostsimulationsofstarformingclustersuseanisothermalequationofstate(EOS).However,thismight
v beanoversimplificationgiventhecomplexinterplaybetweenheatingandcoolingprocessesinmolecularclouds.Theoretical
1 predictionsandobservationssuggestthattheeffectivepolytropicexponentγintheEOSvarieswithdensity.Weaddressthese
5 issues and study the effect of a piecewise polytropic EOS on the formation of stellar clusters in turbulent, self-gravitating
3 molecularcloudsusingthree-dimensional,smoothedparticlehydrodynamicssimulations.Inthesesimulationsstarsformviaa
0 processwecallgravoturbulentfragmentation,i.e.,gravitationalfragmentationofturbulentgas.Toapproximatetheresultsof
1 publishedpredictionsofthethermalbehaviorofcollapsingclouds,weincreasethepolytropicexponentγfrom0.7to1.1ata
4 criticaldensityn ,whichweestimatedtobe2.5×105cm−3.Thechangeofthermodynamicstateatn selectsacharacteristic
0 c c
mass scale for fragmentation M , which we relate to the peak of the observed IMF. A simple scaling argument based on
/ ch
h the Jeans mass M at the critical density n leads to M ∝ n−0.95. We perform simulations with 4.3×104cm−3 < n <
J c ch c c
p 4.3×107cm−3totestthisscalingargument.Oursimulationsqualitativelysupportthishypothesis,butwefindaweakerdensity
- dependenceofM ∝n−0.5±0.1.WealsoinvestigatetheinfluenceofadditionalenvironmentalparametersontheIMF.Weconsider
o ch c
variationsintheturbulentdrivingscheme,andconsistentlyfindM isdecreasingwithincreasingn .Ourinvestigationgenerally
r J c
t supportstheideathatthedistributionofstellarmassesdependsmainlyonthethermodynamicstateofthestar-forminggas.The
s
a thermodynamic state of interstellar gas is a result of the balance between heating and cooling processes, which in turn are
: determinedbyfundamentalatomicandmolecularphysicsandbychemicalabundances.Giventheabundances,thederivation
v
ofacharacteristicstellarmasscanthusbebasedonuniversalquantitiesandconstants.
i
X
r Keywords.stars:formation–methods:numerical–hydrodynamics–turbulence–equationofstate–ISM:clouds
a
1. Introduction conditions,therewouldthusbenoreasonforittobeuniversal.
Thereforeaderivationofthecharacteristicstellarmassthatis
Oneofthefundamentalunsolvedproblemsinastronomyisthe
basedonfundamentalatomicandmolecularphysicswouldbe
origin of the stellar mass spectrum, the so-called initial mass
moreconsistent.
function (IMF). Observations suggest that there is a charac-
teristic mass for stars in the solar vicinity. The IMF peaks at There have been analytical models (Jeans 1902; Larson
this characteristic mass which is typically a few tenths of a 1969; Penston 1969; Low&Lynden-Bell 1976; Shu 1977;
solar mass. The IMF has a nearly power-law form for larger Whitworth&Summers 1985) and numerical investigations
masses and declines rapidly towards smaller masses (Scalo of the effects of various physical processes on collapse
1998;Kroupa2002;Chabrier2003). and fragmentation. These processes include, for exam-
AlthoughtheIMFhasbeenderivedfromvastlydifferentre- ple, magnetic fields (Basu&Mouschovias 1995; Tomisaka
gions,fromthesolarvicinitytodenseclustersofnewlyformed 1996; Gallietal. 2001), feedback from the stars them-
stars,thebasicfeaturesseemtobestrikinglyuniversaltoallde- selves(Silk1995;Nakanoetal.1995;Adams&Fatuzzo1996)
terminations(Kroupa2001). Initialconditionsin star forming and competitive coagulation or accretion (Silk&Takahashi
regionscanvaryconsiderably.IftheIMFdependsontheinitial 1979; Lejeune&Bastien 1986; Price&Podsiadlowski 1995;
Murray&Lin 1996; Bonnelletal. 2001a,b; Durisenetal.
Sendoffprintrequeststo:A.-K.Jappsen,e-mail:[email protected] 2001). In another group of models, initial and environmental
2 A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation
conditions, like the structural properties of molecular clouds InSect.2wereviewwhatiscurrentlyknownaboutthether-
determine the IMF (Elmegreen&Mathieu 1983; Elmegreen mal properties of interstellar gas. In Sect. 3 we approach the
1997a,b,1999,2000b,a,2002).Larson(1973a)andZinnecker fragmentationproblemanalytically,whileinSect. 4weintro-
(1984, 1990) argued in a more statistical approach that the duce our computationalmethod. In Sect. 5 we discuss gravo-
central-limit theorem naturally leads to a log-normal stellar turbulent fragmentation in non-isothermal gas. In Sect. 6 we
massspectrum.Moreover,therearemodelsthatconnectturbu- analyzetheresultingmassdistribution.We furtherinvestigate
lentmotionsinmolecularcloudstotheIMF(e.g.Larson1981; the influenceof differentturbulentdrivingfields and different
Fleck 1982; Padoan 1995; Padoanetal. 1997; Klessenetal. scaleofdrivinginSect.7.Finally,inSect.8wesummarize.
1998,2000;Klessen2001;Padoan&Nordlund2002).Thein-
terplaybetweenturbulentmotionandself-gravityofthecloud
2. ThermalPropertiesofStar-FormingClouds
leads to a process we call gravoturbulent fragmentation. The
supersonicturbulenceubiquitouslyobservedin moleculargas Gravity in galactic molecular clouds is initially expected to
generates strong density fluctuations with gravity taking over be opposed mainly by a combination of supersonic turbu-
inthedensestandmostmassiveregions.Oncegasclumpsbe- lenceandmagneticfields(MacLow&Klessen2004).Theve-
comegravitationallyunstable,collapsesetsin.Thecentralden- locity structure in the clouds is always observed to be dom-
sityincreasesuntilaprotostellarobjectsformandgrowinmass inated by large-scale modes (MacLow&Ossenkopf 2000;
viaaccretionfromtheinfallingenvelope.Formoredetailedre- Ossenkopfetal.2001;Ossenkopf&MacLow2002).Inorder
viewsseeLarson(2003)andMacLow&Klessen(2004). to maintain turbulence for some global dynamical timescales
However,currentresultsaregenerallybasedonmodelsthat andtocompensateforgravitationalcontractionofthecloudas
do not treat thermal physics in detail. Typically, they use a awhole,kineticenergyinputfromexternalsourcesseemstobe
simple equation of state (EOS) which is isothermal with the required.Starformationthentakesplaceinmolecularcloudre-
polytropic exponent γ = 1. The true nature of the EOS re- gionswhicharecharacterizedbylocaldissipationofturbulence
mains a major theoretical problem in understanding the frag- andlossofmagneticflux,eventuallyleavingthermalpressure
mentation properties of molecular clouds. Some calculations asthemainforceresistinggravityinthesmalldenseprestellar
invoke cooling during the collapse (Monaghan&Lattanzio cloudcoresthatactuallybuildupthestars(Klessenetal.2004;
1991; Turneretal. 1995; Whitworthetal. 1995). Others in- Vázquez-Semadenietal.2005).Inagreementwiththisexpec-
clude radiation transport to account for the heating that oc- tation, observed prestellar cores typically show a rough bal-
curs once the cloud reaches densities of n(H ) ≥ 1010cm−3 ance between gravity and thermal pressure (Benson&Myers
2
(Myhill&Kaula1992;Boss1993),orsimplyassumeanadia- 1989; Myersetal. 1991). Therefore the thermal properties of
batic equation of state once that density is exceeded(Bonnell the dense star-formingregionsof molecularcloudsmust play
1994;Bateetal. 1995).Spaans&Silk (2000) showedthatra- animportantroleindetermininghowthesecloudscollapseand
diatively cooling gas can be described by a piecewise poly- fragmentintostars.
tropic EOS, in which the polytropicexponentγ changeswith Early studies of the balance between heating and cooling
gas density ρ. Considering a polytropic EOS is still a rather processes in collapsing clouds predicted temperatures of the
crude approximation. In practice the behaviour of γ may be order of 10K to 20K, tending to be lower at the higher den-
morecomplicatedandimportanteffectslikethetemperatureof sities (e.g., Hayashi&Nakano 1965; Hayashi 1966; Larson
the dust, line-trappingand feedbackfrom newly-formedstars 1969, 1973b). In their dynamical collapse calculations, these
shouldbe takeninto account(Scaloetal. 1998). Nevertheless and other authors approximated this somewhat varying tem-
apolytropicEOSgivesaninsightinthedifferencesthata de- peraturebya simple constantvalue,usuallytakento be10K.
parturefromisothermalityevokes. Nearly all subsequent studies of cloud collapse and fragmen-
Recently Lietal. (2003) conducted a systematic study of tationhaveusedasimilarisothermalapproximation.However,
the effects of a varying polytropic exponent γ on gravotur- thisapproximationisactuallyonlyasomewhatcrudeone,valid
bulent fragmentation. Their results showed that γ determines onlytoafactorof2,sincethetemperatureispredictedtovary
how strongly self-gravitating gas fragments. They found that by this much above and below the usually assumed constant
the degree of fragmentation decreases with increasing poly- value of 10K. Given the strong sensitivity of the results of
tropicexponentγintherange0.2<γ <1.4althoughthetotal fragmentationsimulations like those of Lietal. (2003) to the
amountof mass in collapsedcores appearsto remainroughly assumedequationofstateofthegas,temperaturevariationsof
consistent through this range. These findings suggest that the thismagnitudemaybeimportantforquantitativepredictionsof
IMF might be quite sensitive to the thermal physics. Earlier, stellarmassesandtheIMF.
one-dimensional simulations by Passot&Vázquez-Semadeni As can be seen in Fig. 2 of Larson (1985), observa-
(1998)alreadyshowedthatthedensityprobabilitydistribution tionaland theoreticalstudiesof the thermalpropertiesof col-
ofsupersonicturbulentgasdisplaysadependenceonthepoly- lapsing clouds both indicate that at densities below about
tropic exponentγ. However in both computations, γ was left 10−18gcm−3, roughly corresponding to a number density of
strictlyconstantineachcase.Inthisstudyweextendprevious n = 2.5×105cm−3, the temperaturegenerallydecreaseswith
workbyusingapiecewisepolytropicequationofstatechang- increasing density. In this low-density regime, clouds are ex-
ing γ at some chosendensity. We investigateif a changein γ ternally heated by cosmic rays or photoelectric heating, and
determinesthe characteristic mass of the gasclump spectrum they are cooled mainly by the collisional excitation of low-
andthus,possibly,theturn-overmassoftheIMF. lying levels of C+ ions and O atoms; the strong dependence
A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation 3
ofthe coolingrate ondensitythen yieldsan equilibriumtem- 3. AnalyticalApproach
perature that decreases with increasing density. The work of
Followingtheaboveconsiderations,weuseapolytropicequa-
Koyama&Inutsuka(2000), which assumes thatphotoelectric
tionofstatetodescribethethermalstateofthegasinourmod-
heatingdominates,ratherthancosmicrayheatingashadbeen
elswithapolytropicexponentthatchangesatacertaincritical
assumed in earlier work, predicts a very similar trend of de-
densityρ fromγ toγ :
creasing temperature with increasing density at low densi- c 1 2
ties.Theresultingtemperature-densityrelationcanbeapproxi- P = K1ργ1 ρ≤ρc
cmoartreedspboynadspotowearplaowlytwroitphicaneqexupatoinoennotfofstaabteouwti−th0.γ27=5,0w.7h2ic5h. P = K2ργ2 ρ>ρc (1)
The observationalresults of Myers(1978) shown in Fig. 2 of where K andK areconstants,andP,andρarethermalpres-
1 2
Larson (1985) suggest temperatures rising again toward the sureandgasdensity.Foranidealgas,theequationofstateis:
high end of this low-density regime, but those measurements
k
refermainlytorelativelymassiveandwarmcloudcoresandnot P= B ρT(ρ) (2)
µm
to the small, dense, cold cores in which low-mass stars form. p
AsreviewedbyEvans(1999),thetemperaturesofthesecores where T is the temperature,and k , µ, and m are Boltzmann
B p
aretypicallyonlyabout8.5Katadensityof10−19gcm−3,con- constant, molecular weight, and protonmass. So the constant
sistentwithacontinuationofthedecreasingtrendnotedabove Kcanbewrittenas:
andwiththecontinuingvalidityofapolytropicapproximation
k
withγ≈0.725uptoadensityofatleast10−19gcm−3. K = B ρ1−γT(ρ) (3)
µm
p
SinceKisdefinedasaconstantinρ,itfollowsforT:
T1 = a1ργ1−1 ρ≤ρc
At densities higherthan this, star-formingcloudcoresbe- T2 = a2ργ2−1 ρ>ρc (4)
come opaque to the heating and cooling radiation that de-
wherea anda areconstants.Theinitialconditionsdefinea :
termines their temperatures at lower densities, and at densi- 1 2 1
ties above 10−18gcm−3 the gas becomes thermally coupled a =T ρ1−γ1 (5)
1 0 0
to the dust grains, which then control the temperature by
theirfar-infraredthermalemission.Inthishigh-densityregime, Atρc itholdsthat:
dominated thermally by the dust, there are few direct tem-
T (ρ )=T (ρ ) (6)
1 c 2 c
perature measurements because the molecules normally ob-
served freeze out onto the dust grains, but most of the avail- Thus,a canbewrittenintermsofa :
2 1
able theoretical predictions are in good agreement concern-
a =a ργ1−γ2 (7)
ing the expected thermal behavior of the gas (Larson 1973b; 2 1 c
Low&Lynden-Bell 1976; Masunaga&Inutsuka 2000). The
According to the analytical work by Jeans (1902) on the
balancebetweencompressionalheatingandthermalcoolingby
stability of a self-gravitating, isothermal medium the oscilla-
dustresultsinatemperaturethatincreasesslowlywithincreas-
tionfrequencyωandthewavenumberkofsmallperturbations
ingdensity,andtheresultingtemperature-densityrelationcan
satisfythedispersionrelation
be approximated by a power law with an exponent of about
0.075, which correspondsto γ = 1.075. Taking these values, ω2−c2sk2+4πGρ0 =0 , (8)
thetemperatureispredictedtoreachaminimumof5Katthe
wherec isthesoundspeed,Gthegravitationalconstant,andρ
transitionbetweenthelow-densityandthehigh-densityregime s 0
thegasdensity.Theperturbationisunstableifthewavelengthλ
at about 2 × 10−18gcm−3, at which point the Jeans mass is
exceedstheJeanslengthλ =2π/k or,equivalently,ifthemass
about 0.3M (see also, Larson 2005). The actual minimum J J
⊙ exceedstheJeansmass
temperature reached is somewhat uncertain because observa-
tionshavenotyetconfirmedthepredictedverylowvalues,but 4π λ 3 π5/2
M = ρ J = G−3/2ρ −1/2c3 . (9)
such cold gas would be very difficult to observe; various ef- J 3 0(cid:18) 2(cid:19) 6 0 s
fortstomodeltheobservationshavesuggestedcentraltemper-
Note thatwe define the Jeans mass M as the mass originally
J
aturesbetween6Kand10Kforthedensestobservedprestellar
containedwithinasphereofdiameterλ .
cores, whose peak densities may approach10−17gcm−3 (e.g., J
InasystemwithapolytropicEOS,i.e.,P= Kργ,thesound
Zucconietal. 2001; Evansetal. 2001; Tafallaetal. 2004). A
speedis
power-law approximation to the equation of state with γ ≈
1.075 is expected to remain valid up to a density of about dP 1/2
10−13gcm−3, above which increasing opacity to the thermal cs = dρ! =(Kγ)1/2ρ(γ−1)/2 . (10)
emission fromthe dustcausesthe temperatureto beginrising
Thus,theJeansmasscanbewrittenas
muchmorerapidly,resultinginan"opacitylimit"onfragmen-
tation that is somewhat below 0.01M⊙ (Low&Lynden-Bell π5/2 K 3/2
1976;Masunaga&Inutsuka2000). MJ = γ3/2ρ(3/2)γ−2 . (11)
6 (cid:18)G(cid:19)
4 A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation
Table1.Sampleparameters,nameoftheenvironmentusedin
thetext,drivingscalek,criticaldensityn ,numberofSPHpar-
c
ticles,numberN ofprotostellarobjects(i.e.,“sinkparticles”in
thecentersofprotostellarcores)atfinalstageofthesimulation,
percentageofaccretedmassatfinalstageM /M
acc tot
Name k log n particle N Macc
10 c Mtot
cm−3 number [%]
Ik2 1..2 h —i 205379 59 56
Ik2b 1..2 — 1000000 73 78
Ik2L 1..2 — 9938375 6 4
R5k2 1..2 4.63 205379 22 73
R5k2b 1..2 4.63 1000000 22 70
R6k2 1..2 5.63 205379 64 93
R6k2b 1..2 5.63 1000000 54 61
R7k2 1..2 6.63 205379 122 84
R7k2b 1..2 6.63 1000000 131 72
R7k2L 1..2 6.63 1953125 143 46
R8k2 1..2 7.63 205379 194 78
R8k2b 1..2 7.63 1000000 234 53
R8k2L 1..2 7.63 5177717 309 29
R5k8 7..8 4.63 205379 1 64
R6k8 7..8 5.63 205379 38 68
Fig.1.LocalJeansmassasafunctionofdensityforfourruns
R7k8 7..8 6.63 205379 99 60
with differentcritical densitiesn . For comparisonthe depen-
c R8k8 7..8 7.63 205379 118 72
denceisalso shownforthe isothermalcase (dottedline). The
R5k2r1 1..2 4.63 205379 16 62
Jeansmasschangesatthecriticaldensity.Theinitialmeanden- R6k2r1 1..2 5.63 205379 34 72
sityandthedensityatwhichsinkparticlesformarerepresented R7k2r1 1..2 6.63 205379 111 68
bytheverticaldashedlines.Thedashed-dottedlinesshowthe R8k2r1 1..2 7.63 205379 149 64
minimalresolvablemass for the runs with the highestresolu- R5k2r2 1..2 4.63 205379 21 72
tion. R6k2r2 1..2 5.63 205379 51 74
R7k2r2 1..2 6.63 205379 119 70
R8k2r2 1..2 7.63 205379 184 70
UsingEqs.3,4,5,7and11onefinds: R5k2r3 1..2 4.63 205379 18 90
R6k2r3 1..2 5.63 205379 52 85
MJ1 = π56/2 kBGTµ0ρm10−γ13/2γ13/2ρ(3/2)γ1−2 ρ≤ρc RR78kk22rr33 11....22 67..6633 220055337799 112936 7761
MJ2 = π56/2 kBGTµ0ρm10p−γ13/2γ23/2ρc(3/2)(γ1−γ2)ρ(3/2)γ2−2 ρ>ρc densWityhaρth?aOpnpeenssugwgheesntioγninisctrheaatsethseaibnocvreeausneitiynγatisthseucffirictiiecnalt
 p  tostronglcyreducefragmentationathigherdensities,introduc-
Thesoundspeedchangeswhenthepolytropicindexchangesat ing a characteristic scale into the mass spectrum at the value
ρc,soMJ alsovaries(seeFig.1),suchthat: of the Jeans mass at ρc. Then the behavior of the Jeans mass
withincreasingcriticaldensitywouldimmediatelyallowusto
3
MJ1 = γ1 2 . (12) derivethescalinglaw
MJ2 γ2! M ∝ρ−0.95. (13)
ch c
Ifweuseγ1=0.7andγ2=1.1asjustifiedinSect.2then MJ1 ∝ This simple analytical consideration would then predict a
ρ−0.95. characteristic mass scale which corresponds to a peak of the
Duringtheinitialphaseofcollapse,theturbulentflowpro- IMFat 0.35 M for a criticaldensity of ρ = 10−18gcm−3 or
⊙ c
ducesstrongrampressuregradientsthatformdensityenhance- equivalently a number density of n = 2.5× 105cm−3 when
c
ments. Higher density leads to smaller local Jeans masses, so usingameanmolecularweightµ = 2.36appropriateforsolar
theseregionsbegintocollapseandfragment.Simulationswith metallicitymolecularcloudsintheMilkyWay.Note,however,
an SPH code different from the one used in the present work thatthissimplescalinglawdoesnottakeanyfurtherdynamical
show that fragmentation occurs more efficiently for smaller processesintoaccount.
values of γ, and less efficiently for γ > 1, cutting off en-
tirely at γ > 1.4 (Lietal. 2003; Arcoragietal. 1991). For
4. NumericalMethod
filamentary systems, fragmentation already stops for γ > 1
(Kawachi&Hanawa1998).Thispointisdiscussedinmorede- InordertotestthescalingofthecharacteristicmassM given
ch
tailinSec.5. by Eq. 13, we carry out simulations of regions in turbulent
A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation 5
mance. Introducing sink particles allows us to follow the dy-
namical evolution of the system over many free-fall times.
Once the density contrast in the center of a collapsing cloud
coreexceedsavalueof5000,correspondingtoathresholdden-
sityofaboutn = 4×108cm−3 (seeFig.1),theentirecentral
th
regionisreplacedbya“sinkparticle”(Bate,Bonnell,&Price
1995).Itisasingle,non-gaseous,massiveparticlethatonlyin-
teractswithnormalSPHparticlesviagravity.Gasparticlesthat
comewithinacertainradiusofthesinkparticle,theaccretion
radiusr , are accretedif they are boundto the sink particle.
acc
This allows us to keep track of the total mass, the linear and
angularmomentumofthecollapsinggas.
Each sink particle defines a control volume with a fixed
radius of 310 AU. This radius is chosen such that we always
resolve the Jeans scale below the threshold density n , fol-
th
lowing Bate&Burkert (1997). We cannot resolve the subse-
quentevolutioninitsinterior.Combinationwithadetailedone-
dimensional implicit radiation hydrodynamic method shows
thataprotostarformsintheverycenterabout103yraftersink
creation(Wuchterl&Klessen2001).Wesubsequentlycallthe
sink protostellar object or simply protostar. Altogether, the
Fig.2.Temperatureasafunctionofdensityforfourrunswith
sinkparticlerepresentsonlytheinnermost,highest-densitypart
different critical densities n . The dotted lines show the ini-
c ofalargercollapsingregion.Thetechnicaldetailsontheimple-
tialconditions.Thecurvehasadiscontinuousderivativeatthe
mentationofsinkparticlesintheparallelSPHcodeGADGET
criticaldensity.
canbefoundinAppendixA.
Protostellar collapse is accompanied by a substantial loss
ofspecificangularmomentum,evenintheabsenceofmagnetic
molecularcloudsin which we varythe critical densityρ and
c
fields(Jappsen&Klessen 2004).Still, mostofthematterthat
determinetheresultingmassspectraofprotostellarobjects,and
fallsinwillassembleinaprotostellardisk.Itisthentransported
thus M . Duringgravoturbulentfragmentationitis necessary
ch
inward by viscous and possibly gravitational torques (e.g.,
to follow the gas overseveralordersof magnitudein density.
Bodenheimer1995;Papaloizou&Lin1995;Lin&Papaloizou
Themethodofchoicethereforeissmoothedparticlehydrody-
1996).WithtypicaldisksizesoforderofseveralhundredAU,
namics(SPH). Excellentoverviewsof the method,its numer-
thecontrolvolumefullyenclosesbothstaranddisk.Iflowan-
icalimplementation,andsomeof itsapplicationsare givenin
gular momentum material is accreted, the disk is stable and
reviewsbyBenz(1990)andMonaghan(1992).Weusethepar-
most of the material ends up in the central star. In this case,
allelcodeGADGET,designedbySpringeletal.(2001).SPHis
the disk simply acts as a buffer and smooths eventual accre-
aLagrangianmethod,wherethefluidisrepresentedbyanen-
tion spikes. It will not delay or prevent the mass growth of
sembleofparticles,andflowquantitiesareobtainedbyaverag-
the central star by much. However, if material that falls into
ingoveranappropriatesubsetofSPHparticles.Weuseaspher-
the control volume carries large specific angular momentum,
ically symmetric cubic spline function to define the smooth-
then the massload ontothe disk is likely to be faster than in-
ing kernel (Monaghan1992). The smoothing length can vary
wardtransport.Thediskgrowslargeandmaybecomegravita-
in space and time, such that the number of considered neigh-
tionallyunstableandfragment.Thismayleadtotheformation
borsisalwaysapproximately40.Themethodisabletoresolve
ofabinaryorhigher-ordermultiple(Bodenheimeretal.2000;
large density contrasts as particles are free to move, and so
Fromangetal.2002).
naturally the particle concentration increases in high-density
regions. We use the Bate&Burkert (1997) criterion to deter-
minetheresolutionlimitofourcalculations.Itisadequatefor
4.2.Modelparameters
theproblemconsideredhere,wherewefollowtheevolutionof
highlynonlineardensityfluctuationscreatedbysupersonictur- We include turbulence in our version of the code that is
bulence.Wehaveperformedaresolutionstudywithupto107 drivenuniformlywiththemethoddescribedbyMacLowetal.
SPHparticlestoconfirmthisresult. (1998) and MacLow (1999). The observed turbulent veloc-
ity field in molecular clouds will decay in a crossing time
if not continuously replenished (MacLow&Klessen 2004;
4.1.Sinkparticles
Scalo&Elmegreen2004;Elmegreen&Scalo2004).Ifthetur-
SPHsimulationsofcollapsingregionsbecomeslowerasmore bulencedecays,thenonlythermalpressurepreventsglobalcol-
particlesmovetohigherdensityregionsandhencehavesmall lapse. In our case, we examine regionsglobally supportedby
timesteps. Replacing dense cores by one single particle leads the turbulence at the initial time. We choose to model con-
to considerable increase of the overall computational perfor- tinuously driven turbulence leading to inefficient star forma-
6 A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation
Fig.3. Column density distribution of the gas and location of identified protostellar objects (black circles) using the high-
resolutionmodelsR6..8k2batthestagewhereapproximately50%ofthegasisaccreted.Projectionsinthexy-,xz-,andyz-plane
areshownforthreedifferentcriticaldensities.
tionratherthanagloballycollapsingregionproducingefficient the order of the size of our computational domain, i.e. with
star formation (MacLow&Klessen 2004). This is achieved wavenumbersk = 1..2.Intheadoptedintegrationscheme,we
here by applying a nonlocal driving scheme that inserts en- add the turbulent energy ∆E = E˙ ∆t to the system at each
in in
ergy in a limited range of wave numbers k. MacLow (1999) timestep∆twhenevermorethan40%oftheSPHparticlesare
showsthathydrodynamicalturbulencedecayswitha constant advanced.
energy-dissipationcoefficient. Thus a constant kinetic energy
inputrateE˙ isabletomaintaintheobservedturbulenceandto
in
stabilizethesystemagainstgravitationalcontractiononglobal Inallourmodelsweadoptaninitialtemperatureof11.4K
scales.Thedrivingstrengthisadjustedtoyieldaconstanttur- corresponding to a sound speed cs = 0.2kms−1, a molecu-
bulentrmsMachnumberM =3.2.Furthermore,itisknown lar weight µ of 2.36 and an initial number density of n =
rms
thatthevelocitystructureinmolecularcloudsisalwaysdomi- 8.4 × 104cm−3, which is typical for star-forming molecular
nated by the largest-scale mode (e.g., MacLow&Ossenkopf cloud regions (e.g., ρ-Ophiuchi, see Motte et al. 1998 or the
2000; Ossenkopfetal. 2001; Ossenkopf&MacLow 2002; central region of the Orion Nebula Cluster, see Hillenbrand
Bruntetal.2004),consequentlywe insertenergyonscales of 1997; Hillenbrand & Hartmann 1998). Our simulation cube
holds a mass of 120M and has a size of L = 0.29pc. The
⊙
A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation 7
cubeissubjecttoperiodicboundaryconditionsineverydirec-
tion.ThemeaninitialJeansmassishM i =0.7M .
J i ⊙
WeusetheEOSdescribedinSect.3,andcomputemodels
with4.3×104cm−3 ≤n ≤4.3×107cm−3.Note,thatthelowest
c
and the highest of these critical densities represent rather ex-
tremecases.FromFig.2,whereweshowthetemperatureasa
functionofnumberdensity,itisevidentthattheyresultintem-
peraturesthataretoohighortoolowcomparedtoobservations
andtheoreticalpredictions.Nevertheless,includingthesecases
clarifiesthe existenceofa realtrendin the dependenceofthe
characteristic mass scale on the critical density. Each simula-
tionstartswithauniformdensity.Drivingbeginsimmediately,
whileself-gravityisturnedonatt = 2.0t ,afterturbulenceis
ff
fullyestablished.Theglobalfree-falltimescaleist ≈ 105yr.
ff
Our models are named mnemonically. R5 up to R8 stand for
thecriticaldensityn (4.3×104cm−3 ≤ n ≤ 4.3×107cm−3)
c c
in the equation of state, k2 or k8 stand for the wave numbers
(k = 1..2 or k = 7..8) at which the drivingenergy is injected
intothesystemandbflagstherunswith1milliongasparticles.
TheletterLmarksthehighresolutionrunsforcriticaldensities
n =4.3×106cm−3andn =4.3×107cm−3with2millionand
c c
5.2millionparticles,respectively.Differentrealizationsofthe
turbulentvelocityfieldaredenotedbyr1,r2,r3.Forcompari-
sonwealsorunisothermalsimulationsmarkedwiththeletterI
thathaveparticlenumbersofapproximately200000,1million
and10milliongasparticles.
The number of particles determines the minimal resolv-
ableJeansmassinourmodels.Figure1showsthedependence
of the local Jeans mass on the density for the adopted poly-
tropicequationofstate.Atthecriticaldensitythedependency
of the Jeans mass on density changes its behavior. The mini-
mum Jeans mass M that needs to be resolved occurs at the
res
densityatwhichsinkparticlesareformed.AlocalJeansmass
is considered resolved if it contains at least 2 × N = 80
neigh
SPHparticles(Bate&Burkert1997).AscanbeseeninFig.1 Fig.4. Temporal evolution of the number of protostellar ob-
weareabletoresolve M with1millionparticlesforcritical jects(upperplot)andoftheratioofaccretedgasmasstototal
res
densitiesupton = 4.3×105cm−3.Since thisisnotthecase gasmass(lowerplot)formodelsR5..8k2b.Thelegendshows
c
for n = 4.3×106cm−3 and n = 4.3×107cm−3, we repeat thelogarithmsoftherespectivenumberdensitiesincm−3.The
c c
these simulationswith 2 million and 5.2 million particles, re- times are given in units of a free-fall time τ . We also show
ff
spectively. Due to long calculation times we follow the latter the models R7k2L and R8k2L with 2×106 and 5×106 par-
onlytothepointintimewhereabout30%ofthegashasbeen ticles, respectively, which are denoted by the letter “L”. For
accreted. comparisonthedottedlinesindicatethevaluesfortheisother-
Atthedensitywhereγchangesfrombelowunitytoabove malmodelIk2b.
unity, the temperature reaches a minimum. This is reflected
in the "V"-shape shown in Figure 2. All our simulations start
randomnumberstogeneratethefield whilekeepingthe over-
with the same initialconditionsin temperatureanddensity as
allstatistical propertiesthe same. Thisallowsus toassess the
markedbythedottedlines.
statistical reliability of our results. These models are labeled
In a furtherset ofsimulationswe analyzethe influenceof
fromR5..8k2r1toR5..8k2r3.Second,drivingintwo different
changingtheturbulentdrivingschemeonfragmentationwhile
wavenumberrangesisconsidered.Mostmodelsaredrivenon
usingapolytropicequationofstate.Thesemodelscontain2×
105particleseach. largescales(1≤k≤2)butwehaverunasetofmodelsdriven
onsmallscales(7≤k≤8)forcomparison.
Following Bate&Burkert (1997), these runs are not con-
sidered fully resolved for n ≥ 4.3 × 105 at the density of ThemainmodelparametersaresummarizedinTable1.
c
sink particle creation, since M falls below the critical mass
res
of 80 SPH particles. We note, however,that the globalaccre-
5. Gravoturbulentfragmentationinpolytropicgas
tion history is notstrongly affected. First, we study the effect
of different realizationsof the turbulentdriving fields on typ- Turbulence establishes a complex network of interacting
ical masses of protostellar objects. We simply select different shocks,whereconvergingflowsandsheargeneratefilamentsof
8 A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation
highdensity.The interplaybetweengravityandthermalpres- Thefilamentarystructurethatoccursin oursimulationsis
sure determines the further dynamics of the gas. Adopting a visualizedinFig.3.Hereweshowthecolumndensitydistribu-
polytropicEOS(Eq.1),thechoiceofthepolytropicexponent tion ofthe gasandthe distributionofprotostellarobjects.We
playsan importantrole determiningthe fragmentationbehav- display the results for three different critical densities in xy-,
ior.FromEq.11itisevidentthatγ= 4/3constitutesacritical xz- and yz-projection. The volume density is computed from
value.AJeansmassanalysisshowsthatforthree-dimensional theSPHkernelin3Dandthenprojectedalongthethreeprin-
structures M increases with increasing density if γ is above cipalaxes.Figure3showsforallthreecasesaremarkablyfil-
J
4/3. Thus, γ > 4/3 results in the termination of any gravita- amentarystructure.Thesefilamentsdefinethelociwheremost
tionalcollapse. protostellarobjectsform.
Clearly,thechangeofthepolytropicexponentγatacertain
Also, collapse and fragmentation in filaments depend on
critical density influences the number of protostellar objects.
the equation of state. The equilibrium and stability of fil-
If the critical density increases then more protostellar objects
amentary structures has been studied extensively, beginning
formbutthemeanmassdecreases.
with Chandrasekhar&Fermi (1953), and this work has been
We show this quantitatively in Fig. 4. In (a) we compare
reviewed by Larson (1985, 2003). For many types of col-
the number of protostellar objects for different critical densi-
lapse problems, insight into the qualitative behavior of a col-
tiesn forthemodelsR5...8k2b.Therateatwhichnewproto-
lapsing configuration can be gained from similarity solutions c
starsformchangeswith differentn . Modelsthatswitch from
(Larson2003). For the collapse of cylinderswith an assumed c
low γ to high γ at low densities built up protostellar objects
polytropic equation of state solutions have been derived by
morerarelythanmodelsthatchangeγathigherdensities.
Kawachi&Hanawa (1998), and these authors found that the
Figure4bshowstheaccretionhistories(thetimeevolution
existenceofsuchsolutionsdependsontheassumedvalueofγ:
of the combined mass fraction of all protostellar objects) for
similarity solutions exist for γ < 1 but not for γ > 1. These
the modelsR5..8k2b.Accretion starts for all butone case ap-
authorsalsofoundthatforγ <1,thecollapsebecomesslower
proximatelyatthesametime.InmodelR5k2b,γ=1.1already
and slower as γ approachesunity from below, asymptotically
atthemeaninitialdensity,thusγ doesnotchangeduringcol-
coming to a halt when γ = 1. This result shows in a particu-
lapse.Inthiscaseaccretionstartsatalatertime.Thisconfirms
larlyclearwaythatγ = 1is acriticalcase forthe collapseof
thefindingbyLietal.(2003)thataccretionisdelayedforlarge
filaments. Kawachi&Hanawa (1998) suggestedthat the slow
γ. In the other four cases the accretion history is verysimilar
collapsethatispredictedtooccurforγapproachingunitywill
andtheslopeisapproximatelythesameforallmodels.
in reality cause a filament to fragment into clumps, because
Inbothplotswealsoshowtheresultsfromourhighresolu-
thetimescaleforfragmentationthenbecomesshorterthanthe
tionrunsR7k2LandR8k2L.Thesesimulationswith2million
timescaleforcollapsetowardtheaxisofanidealfilament.Ifthe
and 5.2 million particles, respectively, have an accretion his-
effectivevalueofγincreaseswithincreasingdensityasthecol-
torysimilartothetimeevolutionoftheaccretedmassfraction
lapse proceeds, as is expectedfrom the predicted thermal be-
intherunswith1millionparticles.Thenumberofprotostellar
haviordiscussedinSect.2,fragmentationmaythenbeparticu-
objects,however,is largerfortherunswithincreasedparticle
larlyfavoredtooccuratthedensitywhereγapproachesunity.
numbers. Combining our results in these two figures we find
In their numericalstudy Lietal. (2003) found,for a range of
that an environmentwhere γ changes at higher densities pro-
assumedpolytropicequationsofstate,thattheamountoffrag-
ducesmore,butlessmassiveobjects.Thus,themeanmassof
mentation that occurs is indeed very sensitive to the value of
protostellarobjectsdoesindeeddependonthecriticaldensity
thepolytropicexponentγ,especiallyforvaluesofγnearunity
whereγchangesfrom0.7to1.1.
(seealso,Arcoragietal.1991).
The fact that filamentary structure is so prominent in our
6. Dependenceofthecharacteristicmassonthe
results and other simulations of star formation, together with
equationofstate
the factthatmostof the starsin these simulationsformin fil-
aments,suggeststhattheformationandfragmentationoffila- Further insight into how the characteristic stellar mass may
mentsmaybeanimportantmodeofstarformationquitegen- depend on the critical density can be gained from the mass
erally. This is also supported by the fact that many observed spectra of the protostellar objects, which we show in Fig. 5.
star-formingcloudshavefilamentarystructure,andbytheevi- We plot the mass spectra of models R5...6k2b,model R7k2L
dencethatmuchofthestarformationinthesecloudsoccursin andmodelR8k2Latdifferenttimes,whenthefractionofmass
filaments (Schneider&Elmegreen 1979; Larson 1985; Curry accumulatedinprotostellarobjectshasreachedapproximately
2002; Hartmann2002). As we notein Sects. 2 and3 and fol- 10%,30%and50%.Inthetoprowwealsodisplaytheresults
lowing Larson (1973b, 2005), the Jeans mass at the density ofanisothermalrunforcomparison.Weusedthesameinitial
where the temperature reaches a minimum (see Fig. 2), and conditions and parameters in all models shown. Dashed lines
hence,γapproachesunity,ispredictedtobeabout0.3M ,co- indicatethespecificmassresolutionlimits.
⊙
incidentallyclose to the mass at which the stellar IMF peaks. We find closest correspondence with the observed IMF
Thissimilarityisanadditionalhintthatfilamentfragmentation (see, Scalo 1998; Kroupa 2002; Chabrier 2003) for a crit-
withavaryingpolytropicexponentmayplayanimportantrole ical density of 4.3 × 106cm−3 and for stages of accretion
in the origin of the stellar IMF and the characteristic stellar around 30% and above. At high masses, our distribution fol-
mass. lows a Salpeter-like power law. For comparison we indicate
A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation 9
Fig.5.MassspectraofprotostellarobjectsformodelsR5..6k2b,modelR7k2LandmodelR8k2Lat10%,30%and50%oftotal
mass accreted on these protostellar objects. For comparison we also show in the first row the mass spectra of the isothermal
runIk2b.Criticaldensityn ,ratioofaccretedgasmasstototalgasmass M /M andnumberofprotostellarobjectsaregiven
c acc tot
in the plots. The verticalsolid line showsthe positionof the medianmass. The dottedline hasa slope of -1.3andservesas a
referencetotheSalpetervalue(Salpeter1955).Thedashedlineindicatesthemassresolutionlimit.
the Salpeter slope x ≈ 1.3 (Salpeter 1955) where the IMF is The model R5k2b where the change in γ occurs below
defined by dN/dlogm ∝ m−x. For masses about the median the initial mean density, shows a flat distribution with only
massthedistributionexhibitsasmallplateauandthenfallsoff few, but massive protostellar objects. They reach masses up
towardssmallermasses. to 10M and the minimal mass is about 0.3M . All other
⊙ ⊙
modelsbuildupapower-lawtailtowardshighmasses.Thisis
10 A.-K.Jappsenetal.:Massspectrumfromnon-isothermalgravoturbulentfragmentation
due to protostellaraccretionprocesses, as moreand moregas
gets turned into stars (see also, Bonnelletal. 2001b; Klessen
2001; Schmeja&Klessen 2004). The distribution becomes
morepeakedforhighern andthereisashifttolowermasses.
c
Thisisalreadyvisibleinthemassspectrawhentheprotostellar
objectshaveonlyaccreted10%ofthetotalmass.ModelR8k2L
hasminimalandmaximalmassesof0.013M and1.0M ,re-
⊙ ⊙
spectively.There is a gradualshiftin the medianmass (asin-
dicatedbytheverticalline)fromModelR5k2b,with M =
med
2.5M at50%,toModelR8k2L,withM =0.05M at50%.
⊙ med ⊙
Asimilartrendisnoticeableduringallphasesofthemodelevo-
lution.
This change of median mass with critical density n is
c
depicted in Fig. 6. Again, we consider models R5...6k2b,
modelR7k2Land modelR8k2L.Themedianmass decreases
clearlywith increasingcriticaldensityas expected.Aswe re-
Fig.6.Theplotshowsthemedianmassoftheprotostellarob-
solve higher density contrasts the median collapse mass de-
jects overcritical density for modelsR5..6k2b,modelR7k2L
creases. We fit our data with a straight line. The slope takes
andmodelR8k2L.Wedisplayresultsfordifferentratiosofac-
valuesbetween−0.4and−0.6.Thesevaluesarelargerthanthe
cretedgasmasstototalgasmassM /M .Wefitourdatawith
slope−0.95derivedfromthesimplescalinglaw(Eq.13)based acc tot
straightlinesfordifferentstagesofaccretion.Theslopeshave
oncalculatingtheJeansmassM atthecriticaldensityn
J c
the following values: −0.43±0.05 (solid line), −0.52±0.06
Onepossiblereasonforthisdeviationisthefactthatmost
(dashed-dottedline),−0.60±0.07(dashedline).
oftheprotostellarobjectsaremembersoftightgroups.Hence,
theyare subjectto mutualinteractionsandcompetitiveaccre-
tionthatmaychangetheenvironmentalcontextforindividual natureofturbulentflows.Afurtherdiscussiononthisissuecan
protostars.Thisinturninfluencesthemassdistributionandits befoundinKlessenetal.(2000)andHeitschetal.(2001).
characteristics(see,e.g., Bonnelletal.2001a,b).Anotherpos-
sible reasonis that the mass that goesinto filamentsand then
7.2.Dependenceonthescaleofturbulentdriving
intocollapsemaydependonfurtherenvironmentalparameters,
someofwhichwediscussinSect.7. Figure7eshowstheresultsformodelswherethedrivingscale
has been changed to a lower value (k = 7..8). The overall
dependency of M on n is very similar to the cases of
7. Dependenceofthecharacteristicmasson median c
large-scale turbulence. However, we note considerably larger
environmentalparameters
uncertainties in the exact value of the slope. This holds es-
pecially for the phases where 30% and 50% of the total gas
7.1.Dependenceonrealizationoftheturbulent
masshasbeenconvertedintostars.Oneofthereasonsislower
velocityfield
statistics, i.e., for the smallest critical density only one pro-
Wecomparemodelswithdifferentrealizationsoftheturbulent tostellar object forms. Moreover, it has already been noted
driving field in Fig. 7. We fit our data with straight lines for by Lietal. (2003) that driving on small wavelength results
each stage of accretion. Figure 7a shows the results of mod- in less fragmentation. The forming small-scale density struc-
els R5..8k2 which were calculated with the same parameters ture is not so strongly filamentary, compared to the case of
but lower resolution than the models used for Fig. 6. As dis- small-scale driven turbulence. Local differenceshave a larger
cussed in Sec. 5, although the number of protostellar objects influence on the results for driving on small wavelengths.
changes with the number of particles in the simulation, the Nevertheless,formostofthemodelsthemeanmassdecreases
time evolution of the total mass accreted on all protostellar with increasing critical density. Observational evidence sug-
objectsremainssimilar.Thus,lowerresolutionmodelsexhibit gests that real molecular clouds are driven from large scales
thesamegeneraltrendastheirhigh-resolutioncounterpartsand (e.g.Ossenkopf&MacLow2002;Bruntetal.2004).
show thesame globaldependencies.We notice,however,that
the slope of the M -n relation typically is shallower in
median c 8. Summary
the low-resolutionmodels. This can be seen when comparing
Fig. 6 and Fig. 7a, where we used identical turbulent driving UsingSPHsimulationsweinvestigatetheinfluenceofapiece-
fields. wisepolytropicEOSonfragmentationofmolecularclouds.We
For all four different realizations of the turbulent driving study the case where the polytropic index γ changes from a
fieldshowninFig.7a-dweseeacleartrendofdecreasingme- valuebelowunitytooneaboveatacriticaldensityn .Wecon-
c
dianmass M withincreasingn .We concludethat,qual- siderabroadrangeofvaluesofn aroundtherealisticvalueto
median c c
itatively, the M -n relation is independent of the details determinethedependenceofthemassspectrumonn .
median c c
of the turbulentdriving field but, quantitatively,there are sig- Observationalevidencepredictsthatdenseprestellarcloud
nificant variations. This is not surprising given the stochastic cores show rough balance between gravity and thermal pres-